Question

How do you find \[\cos \left( \dfrac{x}{2} \right)\] if \[\cos \left( x \right)=-\dfrac{31}{49}\] using the half-angle identity?

Answer 1

Hint: In order to find the solution of the given question that is to find \[\cos \left( \dfrac{x}{2} \right)\] if \[\cos \left( x \right)=-\dfrac{31}{49}\] using the half-angle identity apply the formula of trigonometric half-angle identity that is \[\cos \left( x \right)=2{{\cos }^{2}}\left( \dfrac{x}{2} \right)-1\] then substitute the given value which is \[\cos \left( x \right)=-\dfrac{31}{49}\] in this formula and find the value of the remaining unknown term in the equation that is \[\cos \left( \dfrac{x}{2} \right)\] .

Complete step by step solution:
According to the question, given value in the question is as follows:
\[\cos \left( x \right)=-\dfrac{31}{49}\]
We have to find the value of \[\cos \left( \dfrac{x}{2} \right)\] using the above value and half-angle identity.
Now we will apply the formula of trigonometric half-angle identity that is \[\cos \left( x \right)=2{{\cos }^{2}}\left( \dfrac{x}{2} \right)-1\] and substitute the given value then we will have:
\[\Rightarrow -\dfrac{31}{49}=2{{\cos }^{2}}\left( \dfrac{x}{2} \right)-1\]
After this add \[1\] to both the sides of the above equation, we will have:
\[\Rightarrow -\dfrac{31}{49}+1=2{{\cos }^{2}}\left( \dfrac{x}{2} \right)-1+1\]
Now simplify the above equation with the help of addition, we will have:
\[\Rightarrow 1-\dfrac{31}{49}=2{{\cos }^{2}}\left( \dfrac{x}{2} \right)\]
After this take the LCM of the terms in the left-hand side of the above equation, we will have:
\[\Rightarrow \dfrac{18}{49}=2{{\cos }^{2}}\left( \dfrac{x}{2} \right)\]
Now divide \[2\] from both sides of the given equation, we will have:
\[\Rightarrow \dfrac{18}{49\times 2}=\dfrac{2{{\cos }^{2}}\left( \dfrac{x}{2} \right)}{2}\]
After this simplify the above equation with the help of division, we will have:
\[\Rightarrow \dfrac{9}{49}={{\cos }^{2}}\left( \dfrac{x}{2} \right)\]
We can rewrite the above equation as follows:
\[\Rightarrow {{\cos }^{2}}\left( \dfrac{x}{2} \right)=\dfrac{9}{49}\]
Now take square root from both the sides of the equation, we will have:
\[\Rightarrow \cos \left( \dfrac{x}{2} \right)=\pm \dfrac{3}{7}\]
Therefore, the values of \[\cos \left( \dfrac{x}{2} \right)\] are \[\dfrac{3}{7}\] and \[-\dfrac{3}{7}\].

Note: Students make mistakes while not considering both the negative and positive value when they take the square root of a number. Students tend to consider the positive root only which leads to an incomplete answer.